Bounded solutions of dynamical systems in Hilbert space

TL;DR

This paper establishes conditions for the existence of bounded solutions to the Schrödinger equation in Hilbert space, utilizing exponential dichotomy and generalized Green's operators to characterize solutions.

Contribution

It provides necessary and sufficient conditions for bounded solutions of the Schrödinger equation in Hilbert space, expanding understanding of solution behavior under exponential dichotomy assumptions.

Findings

Derived conditions for bounded solutions on the entire real axis.

Represented solutions using generalized Green's operator.

Connected exponential dichotomy with solution boundedness.

Abstract

Necessary and sufficient conditions for existence of bounded on the entire real axis solutions of Schrodinger equation are obtained under assumption that the homogeneous equation admits an exponential dichotomy on the semi-axes. Bounded analytical solutions are represented using generalized Green's operator.